Construction of a multi-soliton-like solutions for non-integrable Schrödinger equations with non-trivial far field

Abstract: This article provides a naturel sequel of previous works [6, 4] regarding the stability of travelling waves for a general one-dimensional Schrödinger equation (N LS) with non-zero condition at infinity. The aim of this article is twofold. First, we prove the asymptotic stability of well-prepared chains of dark solitons and secondly, we construct an asymptotic N -soliton-like solution, which is an exact solution of (N LS), the large-time dynamics of which is similar to a decoupled chain of solitons.

• The paper establishes the construction and exponential asymptotic stability of multi-soliton-like solutions to non-integrable defocusing NLS equations with nonzero background.
• It leverages a hydrodynamic formulation and modulation analysis to manage weak soliton interactions and overcome challenges posed by non-integrability.
• The work provides a unified dynamical and variational approach with implications for nonlinear optics, superconductivity, and Bose-Einstein condensation.

Multi-Soliton-Like Solutions for Non-Integrable Schrödinger Equations with Non-Trivial Far Field

Introduction and Context

This paper addresses the construction and asymptotic dynamics of multi-soliton-like solutions for non-integrable one-dimensional nonlinear Schrödinger equations (NLS) with non-vanishing boundary conditions at infinity. The specific focus is on defocusing NLS equations of the form

$$i\partial_t \Psi + \partial_x^2 \Psi + \Psi f(|\Psi|^2) = 0,$$

under the constraint $|\Psi(x)| \to 1$ as $|x| \to \infty$. Such models generalize the well-studied Gross-Pitaevskii equation and are relevant in superconductivity, superfluidity, nonlinear optics, and Bose-Einstein condensation, where the presence of a nontrivial far field encodes persistent background flow.

The soliton resolution conjecture, which predicts that large-time dynamics decompose into a finite chain of decoupled solitons plus dispersive radiation, has received substantial confirmation for integrable systems via inverse scattering. However, similar structural results for non-integrable models, particularly with non-zero boundary data, are scarce. This work presents a rigorous construction of exact $N$-soliton-like solutions to general (non-integrable) defocusing NLS equations, providing quantitative control on their mutual interactions and large-time asymptotics, and establishing sharp asymptotic stability results within the physically relevant energy space.

Main Results and Technical Architecture

The paper establishes two principal results:

1. Asymptotic stability of well-prepared chains of dark solitons: For a class of defocusing nonlinearities satisfying mild spectral and growth assumptions, initial data close to a superposition of $N$ well-separated, ordered traveling waves yields global evolution converging (in a weak topology appropriate for the energy space) to a chain of $N$ decoupled traveling waves, with explicit exponential relaxation of the parameters to their asymptotic values.
2. Existence and construction of pure multi-soliton-like solutions: The existence of exact solutions realizing the multi-soliton asymptotic profile with exponential convergence is demonstrated for arbitrary admissible velocities and positions. This is achieved despite the absence of integrability and relies on a non-perturbative, variational and dynamical approach, adapted from inverse scattering-free techniques developed for KdV and NLS on zero background.

These results are achieved by constructing the problem in a hydrodynamic formulation (Madelung transform) and utilizing a detailed modulation analysis. The approach requires careful handling of technical obstacles such as the lack of strong continuity in the original variables due to the non-zero background, and the fact that mutual soliton interactions are weak but non-negligible in the non-integrable regime.

Analytical Framework and Well-Posedness

The foundational Cauchy theory is established in the energy space, using Zhidkov-type functional spaces to accommodate the modulational instability of the background and to enforce non-vanishing conditions at infinity. Conservation of energy and local well-posedness are shown for nonlinearities $f \in C^3(\mathbb{R}_+)$ satisfying coercivity and spectral gap conditions.

A key element is the hydrodynamic reformulation, with variables $(\eta, v)$ representing the deviation of the density and the phase gradient from their background values. This transformation is necessary both for technical estimates and to exploit the monotonicity structures in the dynamics of the perturbation.

Soliton Profiles, Modulation, and Chains

Traveling wave solutions (dark solitons) are characterized as minimizers of the Hamiltonian constrained to fixed momentum and constructed for speeds $c \in (c_{\min}, c_s)$, where $c_s$ is the appropriate sound speed. The main theorems require the traveling waves to be spectrally and orbitally stable, ensured by suitable constraints on the nonlinearity.

A well-prepared multi-soliton configuration is then defined as an $|\Psi(x)| \to 1$0-tuple of such solitons with distinct speeds and mutual separation, allowing for exponential decay of their overlap. The bulk of the analysis is devoted to showing that the nonlinearity does not introduce leading-order interactions between well-separated solitons, so that the dynamical evolution essentially tracks the superposition dynamics up to a metastable phase and translation modulation.

Asymptotic Decomposition and Rigidity

A Liouville-type theorem is established, ensuring that any global solution which stays sufficiently close (in energy topology) to a multi-soliton manifold and exhibits suitable decay away from the soliton centers must in fact be an exact multi-soliton configuration, up to global symmetries. This rigidity result is crucial for reducing the study of the large time dynamics to the analysis of parameter flows on the $|\Psi(x)| \to 1$1-soliton manifold.

To demonstrate asymptotic stability, monotonicity formulas for momentum-like quantities and energy are developed, controlling the evolution of localized functionals. These, combined with energy coercivity estimates, yield the exponential decay (in a weak sense) of the perturbation between the solitons and establish the orbital and asymptotic stability.

Exact Multi-Soliton-Like Solutions Construction

The paper adapts the Martel-Merle concentration-compactness strategy (well-established for KdV and zero-background NLS) to this setting. The method involves backward-in-time construction: starting from an exact, decoupled $|\Psi(x)| \to 1$2-soliton at large final time and evolving it via the NLS dynamics, then passing to the limit as initial time recedes to $|\Psi(x)| \to 1$3. Uniform-in-time estimates and a compactness argument produce the desired solution, after careful control of non-decaying background contributions.

The construction demonstrates the persistence of multi-soliton structure in non-integrable, non-zero boundary NLS and provides exponential-in-time decay of the perturbation, i.e., for the asymptotic $|\Psi(x)| \to 1$4-soliton solution $|\Psi(x)| \to 1$5:

$|\Psi(x)| \to 1$6

for $|\Psi(x)| \to 1$7 large and some $|\Psi(x)| \to 1$8.

Implications and Perspectives

These results supply comprehensive analytical tools for the dynamical study of NLS-type flows with persistent backgrounds, closing longstanding methodological gaps between the integrable and non-integrable cases in the context of non-zero boundary conditions. The absence of inverse scattering machinery or Lax pairs in the non-integrable case is overcome via dynamical and variational techniques robust under general perturbations.

On the applied side, these findings are essential for understanding the long-time and interaction dynamics of coherent structures in nonlinear optics and superfluidity, particularly for physically realistic models where nonlinearity does not match the integrable prototype. The construction and quantitative description of exact multi-soliton-like solutions informs both theoretical frameworks and numerical simulations of such phenomena.

From a broader mathematical perspective, the methods and stability architecture developed here provide a blueprint for similar constructions in other non-integrable Hamiltonian PDEs with coherent structures and persistent backgrounds, including other nonlinear dispersive or wave systems.

Conclusion

The paper rigorously constructs and characterizes $|\Psi(x)| \to 1$9-soliton-like solutions for non-integrable 1D defocusing NLS with non-trivial far field, establishing exponential asymptotic stability of chains of dark solitons and providing exact global-in-time solutions realizing the soliton asymptotics. The analysis unifies and extends techniques from integrable and non-integrable settings, introducing robust tools for the study of coherent structure dynamics in nonlinear dispersive PDEs with persistent backgrounds (2603.29906).